The set of sums of two squares plays a significant role in number theory. We establish the existence of several rich monochromatic configurations in the natural numbers by exploiting algebraic structures induced by the set of sums of two squares. The proofs rely on algebraic properties arising from the induced structures on the Stone–Čech compactification of the natural numbers.

For an integer $m>1$, we denote by $P(m)$ the largest prime divisor of m. We improve a result of Stewart [‘On the greatest and least prime factors of ${n!}+1$, II’, Publ. Math. Debrecen 65(3–4) (2004), 461–480] by showing that $\limsup _{n \rightarrow \infty } P({n!}+1)/n \geqslant 1+9\log 2$. More generally, for any nonzero polynomial $f(X)$ with integer coefficients, we show that $\limsup _{n \rightarrow \infty } P({n!}+f(n))/n \geqslant 1+9\log 2$. This improves a result of Luca and Shparlinski [‘Prime divisors of shifted factorials’, Bull. Lond. Math. Soc. 37(6) (2005), 809–817]. These improvements stem from an additional combinatorial idea that builds upon the works mentioned above.

Recently, there has been a large number of works on bilinear sums with Kloosterman sums and on sums of Kloosterman sums twisted by arithmetic functions. Motivated by these, we consider several related new questions about sums of Kloosterman sums parametrised by square-free and smooth integers. Some of our results are presented in the much more general setting of trace functions.

Andrews and El Bachraoui [‘On two-color partitions with odd smallest part’, Preprint (2024), arXiv:2410. 14190] recently investigated identities involving two-colour partitions, with particular emphasis on their connection to overpartitions, and posed questions regarding possible companion results. Subsequently, Chen and Zou [‘Combinatorial proofs for two-colour partitions’, Bull. Aust. Math. Soc. 113(1) (2025), to appear] obtained some companion results by employing q-series identities and generating functions. In addition, they presented a combinatorial proof for one of their own results and one of the results of Andrews and El Bachraoui. They posed questions regarding combinatorial proofs of the remaining companion results. In this paper, we provide such proofs.

Let F be a totally real field. Let $\mathsf {A}$ be a simple modular self-dual abelian variety defined over F. We study the growth of the corank of Selmer groups of $\mathsf {A}$ over $\mathbb {Z}_p$-extensions of a complex multiplication (CM) extension of F. We propose an extension of Mazur’s growth number conjecture for elliptic curves to this new setting. We provide evidence supporting an affirmative answer by studying special cases of this problem, generalising previous results on elliptic curves and imaginary quadratic fields.

This paper studies the $4$-ranks of narrow class groups in certain families of quadratic fields. We prove that for any positive integer n, there exists an integer $s_\lambda (n)$ depending on n and the sign $\lambda $ of the fundamental discriminant D, such that for any choice of $s_\lambda (n)$ integers $t_1, \ldots , t_{s_\lambda (n)}$, there are infinitely many D for which the narrow class group of $\mathbb {Q}(\sqrt {D + t_i})$ has $4$-rank bounded by n for all i. This result extends previous work on $3$-ranks to the case of $4$-ranks.

We determine the conditions for the reducibility of some parametrised families of quadratic and cubic polynomials over finite fields, and count the number of irreducible trinomials. The existence of a factorisation of these polynomials plays an important role in studying the finite groups of exceptional Lie types.

A family of $n\times n$ matrices over a field $\mathbb {F}$ is irreducible if it has no common nontrivial invariant subspace, and minimally irreducible if it is irreducible but has no proper irreducible subfamily. If $\mathbb {F}$ is algebraically closed and $n\ge 2$, a minimally irreducible family has at most $2n-1$ elements. We show that for complex $n\times n$ matrices, $n\ge 3$, a family of minimally irreducible (i) matrix units, (ii) rank one projections, (iii) unicellular matrices and (iv) orthoatomic matrices has k elements where respectively (i) $n\le k\le 2n-2$, (ii) $k=n$, (iii) $2\le k\le n-1$ and (iv) $2\le k\le n-1$. All of the values of k in these ranges are attained. If $n=2$, each such minimally irreducible family has $2$ elements.

A classical theorem of Jordan asserts that if a group G acts transitively on a finite set of size at least $2$, then G contains a derangement (a fixed-point free element). Generalisations of Jordan’s theorem have been studied extensively, due in part to their applications in graph theory, number theory and topology. We address a generalisation conjectured recently by Ellis and Harper [‘Orbits of permutation groups with no derangements’, Preprint, 2024, arXiv:2408.16064], which says that if G has exactly two orbits and those orbits have equal length $n \geq 2$, then G contains a derangement. We prove this conjecture in the case where n is a product of two primes, and in the case where $n=bp$ with p a prime and $2b\leq p$. We also verify the conjecture computationally for $n \leq 30$.

Let G be a finite group and p be a prime number. An element g of G is called an $\mathrm {SM}^*$-vanishing element of G if there exists a strongly monolithic character $\chi $ of G satisfying $Z(\chi )=\ker \chi $ and $\chi (g)=0$. In this paper, we present some results on the relationship between the orders of $\mathrm {SM}^*$-vanishing elements of G and the structure of G.

The structure of groups for which certain sets of commutator subgroups are finite is investigated, with a particular focus on the relationship between these groups and those with finite derived subgroup.

We study the restriction of the absolute order on a Coxeter group W to an interval $[1,w]_T$, where $w\in W$ is an involution. We characterise and classify those involutions w for which $[1,w]_T$ is a lattice, using the notion of involutive parabolic subgroups.

An element x of a lattice L is modular if L has no five-element sublattice isomorphic to the pentagon in which x would correspond to the lonely midpoint. We classify all modular elements of the lattice of all monoid varieties.

Let G be a locally compact topological group and $\mathcal {L}(G)$ the space of all its closed subgroups endowed with the Vietoris topology. Let $\mathcal {L}_c(G)$ be the subspace of all compact subgroups of G. Any continuous morphism $\varphi \colon G\to H$ between locally compact groups G and H functorially induces a continuous map $\varphi _*\colon \mathcal {L}_c(G)\to \mathcal {L}_c(H)$ given by $\varphi _*(L)=\varphi (L)$. The main problem addressed in this paper is that of determining the relationship between the openness of $\varphi $ and the openness of $\varphi _*$. For example, we show that if G is locally compact with compact identity component and H is locally compact and totally disconnected, then $\varphi $ is open if and only if $\varphi _*$ is open.

We give a new proof of the singular continuity of Minkowski’s $?$-function. Our proof follows by showing that the maximal Lyapunov exponent of a specific pair of $3\times 3$ nonnegative integer matrices related to Stern’s diatomic sequence is strictly greater than $\log 2$.

From classical trigonometric formulas, complex differential equations of various types have been formulated and widely studied. We investigate perturbed nonlinear complex differential equations and explore their corresponding complex differential systems. Some open questions on nonlinear complex differential equations and systems are proposed.

In this article, we clarify the relation between the squeezing function and the Fridman invariant corresponding to a general domain $\Omega $ (not necessarily convex), where $\Omega $ is defined by

$$ \begin{align*}\Omega = \bigg\lbrace z \in \mathbb{C}^{r_{1}}\times\mathbb{C}^{r_{2}}\times\cdots\times\mathbb{C}^{r_{s}} : \sum\limits_{i\in I_{k}} ||z_{i}||^{m_{i}} < 1, 1\leq k \leq p \bigg\rbrace,\end{align*} $$

with $I_{k}\cap I_{l} = \emptyset $ if $k\neq l$, $I_{1}\cup I_{2} \cup \cdots \cup I_{p} = \lbrace 1, 2, \ldots , s\rbrace $, $n = r_{1} + r_{2} + \cdots + r_{s}$ and $m_{i}> 0$ for all i. Furthermore, we give an example of a domain whose squeezing function corresponding to $\Omega $ is not plurisubharmonic.

In this paper, we give a transformation formula of Dwork’s p-adic hypergeometric function between t and $t^{-1}$. As an appendix, we introduce a finite analogue of this transformation formula, which implies a special case of the transformation formula for Dwork’s p-adic hypergeometric function.

We show that the ‘ternary’ elliptic functions that were introduced and studied by Du Val in 1964 are the $n=3$ instances of n-ary elliptic functions that are defined for arbitrary integers n greater than unity. We also trace the general n-ary elliptic function back to 1886 and the ‘fort remarquable’ function of Halphen.

Recently, Barros et al. [‘On the shortest distance between orbits and the longest common substring problem’, Adv. Math. 344 (2019), 311–339] adopted a dynamical system perspective to study the decay of the shortest distance between orbits. We calculate the Hausdorff dimensions of the exceptional sets arising from the shortest distance between orbits in conformal iterated function systems.